Q:

So how do control engineers control nonlinear plant with exponential output?

Say that a system (circuit, pump, population model, etc.) has a exponential output of the type $$y = ae^{bx}$$
What do control engineers do in order to achieve control objectives such as tracking and stabilization around a set point?

A:

It's usually no problem at all. Take as a simple electronic example an op-amp precision rectifier. It has to cope with severe non-linearities i.e. the diode in the output circuit. It has a formula shown below: -

Where I is the current thru the device and V is the applied voltage to the terminals. It's exponential like in the question but the circuit below copes well: -

Vout has is produced and is perfectly linear with respect to the peak input voltage i.e. negative feedback has overcome the difficulties. The same is ture for a push pull amplifier with negative feedback. You can make one without final transistor biasing (to overcome the distortion arising when changing from one transistor to another) by choosing an op-amp that is high-speed. 

In effect, when the transistors are at the crossover-point the op-amps's high speed almost perfectlyy compensates and "rushes" the input voltage to the transistors past this cross-over point. Again, negative feedback comes to the rescue: -

Real difficulties arise in control systems with complex frequency responses, hysterisis and deadband. Now these can cause major headaches.

