Absolute stability analysis of non‐linear active disturbance rejection control for single‐input–single‐output systems via the circle criterion method

Abstract

This study focuses on the stability analysis of non-linear active disturbance rejection control (ADRC) for singleinput-single-output systems. Firstly, a non-linear ADRC system for a linear plant is transformed into a Lurie system. Secondly, two extended circle criteria are obtained, and two numerical examples are presented to illustrate the absolute stability analysis, including both stable and unstable linear plants. Thirdly, local asymptotic stability of a non-linear ADRC system for a non-linear plant is also performed through linearisation by Taylor expansion. Finally, a comparison with the existed processing methods is further made, including the describing function method and time domain stability analysis method. It can be concluded that the circle criterion method is more convenient and practical for its frequency domain and graphical interpretation. The circle criterion method can also be extended to the stability analysis of a control system which applies linear ADRC to a plant with one non-linear term. IntroductionThe presence of model uncertainties and external disturbances in most industrial control problems put forward a big challenge for modern control theory. The classical proportional-integralderivative (PID) control law is still dominant even today across various sectors of the entire industry for its error driven rather than model-based. However, the active disturbance rejection control (ADRC) systematically proposed by Han in [1, 2] is a drastic departure from both modern and classical control theories. It is a great reflection about 'control theory: model analysis approach or a direct control approach' [3] and 'linear and non-linear of feedback system' [4] and so on. On the one hand, it inherits from merits of PID that it requires little information about the plant. On the other hand, it takes from best offering of modern control theory that it uses an extended state observer (ESO) to estimate and compensate the internal and external disturbances and uncertainties, which is a big breakthrough of 'internal model theory' and 'absolute invariance principle'. Since ADRC was proposed, it has gradually obtained a wide range of practical applications, like parallel active power filters [5], attitude tracking of rigid spacecraft [6], uncertain multivariable systems with time-delay [7], non-linear two-mass drive system [8], the permanent magnet synchronous motor [9], the trajectory tracking control of a flexible-joint robotic system [10], a buck-boost-converter/DC-motor system [11], superconducting radio-frequency cavities [12] to name only a few.The ADRC prefers to use non-linear functions in the design of the observer and the control law, which is potentially much more effective in tolerance to uncertainties and disturbance and improvement of system dynamics. However, it may make the system produce some complex but colourful non-linear behaviours, such as multiple equilibrium points, limit cycles, bifurcations, and chaos. Meanwhile, it is difficult to perform stability and performance analyses, which...

Trajectory tracking control for a quadrotor UAV via extended state observer

“…The estimated disturbances z 3 is included inξ . Thus, the Lurie system is displayed in Figure 5 (Gan & Han, 2003;Li et al, 2015).…”

Control of systems with sector-bounded nonlinearities: robust stability and command effort minimization by disturbance rejection

“…Several stability results are available in the context of disturbance rejection control, see, e.g., [17], [18], [19], but under somewhat different assumptions from here. In this section, a stability result suitable for the framework described in Section II is derived.…”

“…In this section we propose a novel approach for designing the gain matrices K, L x and L d in such a way that stability condition (24) is satisfied. The approach is based on the derivation of high-and low-frequency asymptotic approximations of the transfer functions in (18). The intersections between the high-and low-frequency asymptotes will allow us to find the gains ensuring robust stability.…”

“…Observers of this kind are well known in the literature and are commonly referred to as extended state observers (ESO), disturbance observers, or unknown input observers [1], [3], [17]. Extended observers are at the core of Active Disturbance Rejection Control [4], [18], and of Embedded Model Control [5], [6].…”

“…However, these techniques are affected by relevant problems, such as high sensitivity to measurement noise, [13], [14] and peaking phenomena [7]. Robust stability in the presence of input nonlinearities and uncertainty has been one of the major research subjects of Active Disturbance Rejection Control, but with somewhat different assumptions from here [17], [18], [19].…”