On convergence of extended state observers for nonlinear systems with non-differentiable uncertainties

The analysis of convergence of extended state observers (ESOs) requires the total disturbance to be differentiable. However, this requirement is not satisfied in many control engineering practices. In this paper, we attempt to analyze the convergence of ESOs for nonlinear systems with non-differentiable uncertainties. A decomposition method is first presented to divide the non-differentiable total disturbance into a differentiable signal and a bounded but non-differentiable signal. Based on this decomposition, we give out the convergence of both nonlinear and linear ESOs (NLESO/LESO), low- and high-power ESOs (LPESO/HPESO), and fixed-time ESO (FxESO). We also derive the explicit formulas for the estimation errors of these ESOs. Simulations and experiments demonstrate the correctness of the analysis results.

Control of single input Hamiltonian systems based on the flatness of their tangent linearization

We explore the incremental flatness based control of single input under-actuated nonlinear Hamiltonian systems exhibiting a controllable tangent linearization model around a given equilibrium point. General properties of controllable linearized Hamiltonian systems are presented, which significantly ease the stabilizing, or output reference trajectory tracking, feedback controller design for the nonlinear system. Controllability of the tangent linear system is equivalent to its flatness. A flat filter controller and a set of nested controllers are presented which are based on incremental position measurements alone. There will be no need for explicit estimations of the conjugate momenta through observers. The proposed controller is constituted by a set of nested, second order linear Flat Filter compensation networks acting on the second order pure integration models, naturally present in the input-to-flat output system structure. For pure integration perturbed systems, Flat Filters have been shown to be equivalent, in general, to reduced order extended state observer based Active Disturbance Rejection controllers. A challenging stabilization problem for an unstable, nonlinear, two spring-mass system carrying an inverted pendulum is considered. Computer simulations illustrate the effectiveness of the proposed controller design in the presence of flat output measurement noise.

Fixed-time trajectory following for quadrotors via output feedback

A fixed-time trajectory following problem for quadrotors via output feedback is concerned. Based on the inner-outer separation design philosophy, the under-actuated quadrotor is formulated as a hierarchical structure composed by position and attitude dynamics. With an emphasis on removing the demand on unmeasured velocity and eliminating the negative effect of disturbances, fixed-time extended state observers utilizing two kinds of polynomial feedback terms are proposed to simultaneously identify unavailable velocity states and unknown uncertainties with a fixed-time estimation capability. With these observation results, a velocity free fixed-time control protocol is synthesized to enable a satisfied trajectory regulation with a uniform convergence time independent of initial positions, such that a prescribed fixed-time stability and enhanced robustness can be obtained with chattering-free inputs. By virtue of bi-limit homogeneity properties, all error variables of the resultant quadrotor system are demonstrated to be fixed-time convergent. Eventually, the benefits of developed algorithm are illustrated via simulations.

Robust disturbance rejection control of a biped robotic system using high-order extended state observer

This study addressed the problem of robust control of a biped robot based on disturbance estimation. Active disturbance rejection control was the paradigm used for controlling the biped robot by direct active estimation. A robust controller was developed to implement disturbance cancelation based on a linear extended state observer of high gain class. A robust high-gain scheme was proposed for developing a state estimator of the biped robot despite poor knowledge of the plant and the presence of uncertainties. The estimated states provided by the state estimator were used to implement a feedback controller that was effective in actively rejecting the perturbations as well as forcing the trajectory tracking error to within a small vicinity of the origin. The theoretical convergence of the tracking error was proven using the Lyapunov theory. The controller was implemented by numerical simulations that showed the convergence of the tracking error. A comparison with a high-order sliding-mode-observer-based controller confirmed the superior performance of the controller using the robust observer introduced in this study. Finally, the proposed controller was implemented on an actual biped robot using an embedded hardware-in-the-loop strategy.