I thought they are synonyms. Am I not right?

I think there maybe some differences between them. A "disturbance" is of external, e.g. the affect of wind for the aircraft control system, and the "perturbation" is of internal, e.g. the uncertainties of the parameters for the aircraft control system.

I really appreciate all of you for the valuable comments. 

A disturbance is an external input to your system, while a perturbation is some uncertainty (including parametric and dynamic) in your model.  

@Zigang Pan

I liked your difference between these notions - they are used in practice mostly this way!

Regards!

I think we are confusing the word game. A system is known as a "perturbed system" if it is deviated from its nominal behaviour. The sources that produce such deviation are usually known as perturbations which include 1) any signal external to the system (i.e. disturbance whether matched (input disturbance) or unmatched (internal parametric variations)) and 2) model imperfection (again parametric uncertainty) 3) Any measurement or actuation noise.

I am in line with Imran. In systems thinking, perturbation refers to the deciation of a system away from an 'average' state or equilibrium, which can also indicate system's response to disturbance(s).

Sterman, JD 2000, Business dynamics: Systems thinking and modeling for a complex world, Irwin McGraw-Hill, Boston.

Disturb is to distract, disrupt, etc. Perturb is to disturb and subsequently cause annoyance. Like when someone disturbs your slept, you are then perturbed.

http://painintheenglish.com/case/40

Please refer to these articles:

https://www.researchgate.net/publication/313576646_From_PID_to_Nonlinear_State_Error_Feedback_Controller

https://www.researchgate.net/publication/312046377_Improved_Sliding_Mode_Nonlinear_Extended_State_Observer_based_Active_Disturbance_Rejection_Control_for_Uncertain_Systems_with_Unknown_Total_Disturbance

https://www.researchgate.net/publication/309655805_On_the_Improved_Nonlinear_Tracking_Differentiator_based_Nonlinear_PID_Controller_Design

Article From PID to Nonlinear State Error Feedback Controller

Article Improved Sliding Mode Nonlinear Extended State Observer base...

Article On the Improved Nonlinear Tracking Differentiator based Nonl...

A perturbation is a deviation of the "normal" behaviour of the system. A disturbance is a cause. A perturbation is an effect.

I think, the answer has already been explained very well by all experienced researchers. I agree with the answer of @zigang pan

In mathematics, perturbations are deviation from nominal to an arbitrarily small degree (say epsilon) while disturbance can be of any order in magnitude.

Hence, generally, we assume that the system model is accurate up to some perturbation, while input to the system can be of any magnitude and are described as disturbances if the input causes undesired results.

I am agree with Zigang Pan In other word the disturbance is an external input to the system, which affected on the output response equation multiplied by Sensitivity function in output equations, a perturbation is uncertainty (including parametric and dynamic). 

I guess there is an ambiguity of these terms concerning the context in which they are used because the semantic regarding the cause and effect could be, sometimes, confusing.

In the case of circular causal systems, this difficulty is understandable, but there are a lot of systems containing feedback loops that could be characterized as circularly causal.

Therefore, a disturbance should be regarded as an external input or event and the perturbation as its effect.

Conclusion: disturbances can cause perturbations.

We can possibly adopt sinusoidal disturbance(s); namely when we are aiming at smooth transition(s) (continuous first derivative) when imposing a disturbance to manipulated variable(s) (input) that influences the dynamic system response, as monitored by observed variables (output). That would exclude considering a step impulse or a Dirac delta impulse. Sinusoidal disturbance(s) vanishes at node points. A continuous first derivative may be of great importance to avoid impermissibly high torque surges in mechanical rotating equipment. Sinusoidal disturbance is sometimes helpful while dealing with multicollinearity in system identification.

Application example ― While investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the recursive least squares algorithm (RLS) with forgetting factor (RLS-FF) to estimate the parameters from the KLa correlation, used to predict the O2 gas-liquid mass-transfer, while giving increased weight to most recent data. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The proposed (adaptive) control algorithm compared favourably with PID. Simulations assessed the effect of numerically generated white Gaussian noise (2-sigma truncated) and of first order delay. The power dissipated by agitation was accessed by a torque meter (pilot plant). This investigation was reported at (MSc Thesis):

Thesis Controlo do Oxigénio Dissolvido em Fermentadores para Minimi...

It is from Merriam as follows:

disturbance: a change in the position, arrangement or order of sth;

purturbation: a change in normal state or regular movement of sth. A disturbance of motion, course, arrangement, or state of equilibrium.

So I agree with Ciprian Palaghianu that disturbances can cause perturbations.

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I need to correct and update my answers as the above:

In C.D. Johnson's DAC theory, disturbance “consists of the collection of all system external inputs, internal structure/parameter variations, and 'initial-conditions' which cannot be controlled by the system control designer”

In Perturbation Methond, it is uncertainty factor of one model state's subcomponent（rapid change）. And the general math models between DAC and Perturbation method are different in essence.

Additionally, perturbation has definite mathematical meanings.