Internal Stability Of Low-Order Linear Active Disturbance Rejection Control

Active disturbance rejection control(ADRC)is a new general control method.It combines the essence of classical PID control and modern control ideology,has the superiority of reduced structure,less parameters to be adjusted,and can overcome the dynamics with no model and external disturbance of the systems.Linear active disturbance rejection control further linearizes the parameters of the extended state observer(ESO)and feedback control in the ADRC,which makes the structure simpler and the parameters to be adjusted more concise.It is more suitable for engineering applications,especially for low order plants.In this paper,the internal stability of low order linear active disturbance rejection control is studied,and the stability conclusion is explored in theory and practical application.The details are as follows:Firstly,in the study of the relationship between total disturbance and stability of ADRC,it is often assumed that the total disturbance is bounded,but the rationality and necessity of this assumption are questionable.The total disturbance bounded means that both the internal disturbance and the external disturbance are bounded.It is not reasonable to assume that the internal disturbance is bounded,because the internal disturbance may destroy the stability of the control object,and the internal disturbance itself will diverge after losing the stability.Assuming that the internal disturbance is bounded,the possibility of the system divergence is eliminated,which leads to the stability proof close to the cyclic proof;However,it is unnecessary to assume that the external disturbance is bounded in many cases,because for a typical system such as linear time invariant(LTI)system,the external disturbance will not affect its stability.Considering that there are important differences between the effects of internal disturbance and external disturbance on stability,it is advisable to study them separately.Based on this idea,this paper discusses the constancy of linear active disturbance rejection control(LADRC).For the simple and typical case that the first-order LADRC and the second-order LADRC control the first-order and second-order LTI plants respectively with additive external disturbance,the internal stability is selected as the definition of stability.A necessary and sufficient condition for the internal stability of LADRC system is found in time domain.The condition is related to the uncertain parameters of the object,independent of the external disturbance,and does not limit the size of the internal disturbance.Therefore,the stability of LADRC does not need to be bounded by internal disturbance or external disturbance.Based on this idea,the internal stability of LADRC control for first-order integrator,second-order zero free object and second-order zero with minimum phase is further discussed.Secondly,on the basis of the above work,it is found that the first order LADRC can deal with the object with unbounded derivative of external disturbance.By constructing an example,it is proved that the result in this paper is indeed a necessary and sufficient condition for stability,which is better than some existing literatures.The theoretical results are applied to the stability analysis of the second order LADRC of turbofan engine to ensure the stability of the control scheme.