Electronic – Serial diode to universal motor and brush lenghts differs on intermittent run, why

You do not tamper with the motor. You buy a step-down transformer rated for the current the motor needs. Get the voltage and current ratings of the motor, then just search the web for 'step-down' transformers that match your 230 to 240 vac line voltage on the primary side, and 120vac on the secondary side.

The transformer needs to handle the start-up current as well (it may be listed on the motor), so it maybe rated for twice the motor run current - or more. They are plentiful and have many sources and should not cost much. They are a quick and safe solution.

Usually a simple P,PI loop is better than bad PID loop. You can put 2 PI controlers for speed, for each wheel it own PI controller. Then you just control the speed of each wheel separately. Here is a small pseudo algorithm of PI controller with trap integration method: \$ u(k) = K_p \bigg[ e(k)-e(k-1)+\dfrac{T_s}{2T_i}\big[ e(k)+e(k-1)\big] +\\ +\dfrac{T_d}{T_s}\big[ e(k)-2e(k-1)+e(k-2)\big] \bigg] + u(k-1) \$

EDIT 1: This is a block diagram of PMDC, notice that at the output there is an integrator \$\dfrac{1}{s}\$ that integrates the speed \$\Omega\$ into position \$\Theta\$. It matters if your controller uses speed or velocity feedback (or both). Since the veloctity feedback is obtained derivating the position, that's how is usually done, you can use PI, else a P controller is enough good, but it will give you I like response of the entire loop P->integrating->I. Using PD will give you PI like response PD->interating->PI(IP). You should decide wheather you will use velocity or position as setpoint. Professional equipment uses combined period and frequency measuring for estimating the velocity feedback information from position feedback, so simply derivating the position will give you bad result at low speed.

IMO you should start with making a setpoint velocity generator with ramp, then integrate into position setpoint and use position as setpoint.

\$ v_{set}(k)= v_{set}(k) + \Delta v_{max}\$
\$ p_{set}(k)= p_{set}(k) + v_{set}(k)\cdot T_{sample} \$ integrating position
This integrating position should be exactly the same size as your counter, for example 32-bit integer, so you have to parse it in integer form. When it will roll over, this won't affect the calculation of the error: \$\varepsilon = p_{measured} - p_{set} \$

EDIT 2:

v_max ..maximum speed [pulses/s]
Acc..maximum acceleration [pulses/s^2] or
Acc=v_max/Tramp..Tramp ramp time to max velocity
T_s...sampling time [s]

if (FWD==1) {
   v_set=v_set+Acc*T_s; //pulses per second
   if v_set>v_max v_set=v_max; } 
elsif (BKW==1){ 
   v_set=v_set-Acc*T_s;
   if v_set<-v_max v_set=-v_max;}
else{          //stop moving
   if v_set>0.0 {
      v_set=v_set-Acc*T_s;
      if v_set<0.0  v_set=0.0;
    }
   if v_set<0.0 {
      v_set=v_set+Acc*T_s;
      if v_set>0.0  v_set=0.0;
    }
    }

p_float=p_float + v_set*T_s; //pulses
if p_float>32767.0 p_float=p_float-65535.0; //for 16-bit position counter
else if p_float<-32768.0 p_float=p_float+65535.0;

p_int = int_16(p_float);
epsilon=p_int-p_measured; //all numbers equal int16, 32,...
y_out=Kp*float(epsilon)+v_set; //pulses per second, velocity is fed forward
y_pwm = y_out * k_pulse;//rpm , k_pulse [rpm/pulses]
y_pwm = y_pwm * kv; //volts, kv[V/rpm] from motor spec
y_pwm = y_pwm * k_pwm; //% k_pwm[%/V] - 100% equals to Vcc volts, approx.
if y_pwm>100.0 y_pwm=100.0; // max positive limit pwm DT
else if y_pwm<-100.0 < y_pwm=-100.0; //max neg limit pwm DT

EDIT 3: Industrial PID contrller zn3fd source code. The algorithm is incremental : \$u(k)=u(k-1)+\Delta u(k), \;\Delta u(k)= \Delta P+ \Delta I+\Delta D\$ Transfer function: \$ G(s) = K_p(1+\dfrac{1}{sT_i}+ \dfrac{sT_d}{1+sT_d/\alpha}) \$ Note \$alpha\$ is a filtering factor of D-component it is reccomended to be from 4 to 20, default value is 10. Bigger value means less filtering, lower value means more filter, it makes no sense to use low values.

http://www.mathworks.com/matlabcentral/fx_files/8381/1/content/help/html/STCSL.htm

// industrial PID controller, author: Bobal et al.
// parameters: Kp, Ti, Td, alpha, u_in, u_max
// inputs: Setponit, ProcesVar 
// output: u

double ek, ek1, ek2, uk1, uk2, Setponit, ProcesVar;
double Ts, gamma, u_min, u_max;
double Kpu, Tu, Kp, Ti, Td, Tf, cf, ci, cd;
double q0, q1, q2, p1, p2;

//initialzation, calcualte coefficients
ek1=ek2=uk1=uk2=u=0.0;
Tf = Td/alpha;
cf = Tf/Ts;
ci = Ts/Ti;
cd = Td/Ts;
p1 = -4*cf/(1+2*cf);
p2 = (2*cf-1)/(1+2*cf);
q0 = Kp * (1 + 2*(cf+cd) + (ci/2)*(1+2*cf))/(1+2*cf);
q1 = Kp * (ci/2-4*(cf+cd))/(1+2*cf);
q2 = Kp * (cf*(2-ci) + 2*cd + ci/2 - 1)/(1+2*cf);

// ISR PID algo
ek2 = ek1;
ek1 = ek;
ek = (Setpoint - ProcesVar);
uk2 = uk1;
uk1 = u;

u = q0*ek + q1*ek1 + q2*ek2 - p1*uk1 - p2*uk2;
//limit
if   u>u_max u = u_max;
else if u<u_min u = u_min;